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Civil Engineering Departaieull . | 
UNIVERSITY | OF ILLINOIS, a “ “s 


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Please return to ) ENGINEERING HALL cA a ae 
Room No: 22-2 Gase No, se 


Drawer No. 





Drawing’ Wo, sccm a hs : 


REMOTE STORAGE © 


Property of the . 
oN eis DEPARTMENT uy 





THE UNIVERSITY 


OF ILLINOIS 


LIBRARY 


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UNITED STATES HYDROGRAPHIC OFFICE. 
No. 91. 





dae 3.1 


OF 


MERIDIONAL PARTS, 


TERRESTRIAL SPHEROID, 


i oe Ux ee 
Compt BOT ie ae 


BY 


G. W. LITTLEHALES awnp J. 8. SIEBERT, 


Ur 5- FY DROGRAPHIC OFFICE: 


Under the direction of 
LIEUTENANT G. L. DYER, U. S. N., 
Hydrographer. 





WASHINGTON: 
GOVERNMENT PRINTING OFFICE. 
1889. 


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INTRODUCTION. 


The Mercator projection (named after its originator, Gerhard Kra- 
mer, a German, who, according to the custom of that time, 1569, Latin- 
ized his name) results from the development, upon a plane, of a cylinder, 
tangent to the terrestrial spheroid along the equator, upon which the 
parallels of latitude and meridians of longitude have been projected in 
such a manner that a loxodromic curve, or the track of a ship when the 
course is invariable, shall appear upon the development as a straight 
line. 

The line of tangency, or the equator of the earth, being the circum- 
ference of a right section of the cylinder, will appear as a straight line 
upon the development, while the different elements of the cylinder, cor- 
responding to the series of terrestrial meridians, will appear upon the 
development as a system of equidistant straight lines, parallel to one 
another and perpendicular to the equator, maintaining the same relative 
positions as the corresponding meridians have on the terrestrial spheroid. 
The series of terrestrial parallels will also appear as a system of straight 
lines, paralle] to one another and to the equator, and will so intersect 
the meridian lines as to form a series of rectangles whose successive 
heights must increase in proceeding from the equator in such a manner 
that the required equality of angles shall be preserved for correspond- 
ing elements of the loxodromic curve on the sphere and of its repre- 
sentation on the development. 

The computation of a table of Meridional Parts, or Increased Lati- 
tudes, has for its object to determine the respective distances from the 
equator of the various parallels of latitude, in accordance with the 
ascertained law of progression by which the fundamental condition of 
the Mercator projection shall be realized. 


FORMULA FOR COMPUTING MERIDIONAL PARTS. 


The figure of the earth is very nearly that of an oblate spheroid, that 
is, an ellipsoid generated by the revolution of an ellipse about its minor 
axis. The section made by a plane through the earth’s axis is nearly 
an ellipse, of which the major axis.is the equatorial and the minor axis 
the polar diameter of the earth. Accurate geodetic measurements have 

3 


248735 


a 


Shown that there are small deviations from the regular ellipsoid, but it 
is sufficient to assume all the meridians to be ellipses with the mean 
dimensions deduced from all the measures made in various parts of — 
the earth. 

Referring tothe figures, let Pm’‘cte’P’e’’ represent a meridian of 
the terrestrial spheroid whose polar axis is PP’, and whose equator is 
ee'e/’; let darepresent an element of a loxodromic curve upon the sur- 
face.of the sea between two consecutive meridians me and m/e’; and 
let LO represent the same element on the Mercator chart, as included 


M' mM 
¢ 
T 
E' E 





between the corresponding meridians ME and M’R’. Let lé and LT 
be taken parallel to the elements ee’ and EE’ of the respective equators 
included between the same meridians. Then the condition before men- 
tioned, that the angles of bearing lct and LCT shall be equal, re- 
quires that 

GD sch sleet 


or, since LT=EE’=ee/=the element of the terrestrial equator, it is 
necessary that 


OT :.ct :: ee : lt 


Hence putting 
dm=ct, the meridional element of the terrestrial spheroid, 
adM=CT, the meridional element of the Mercator projection, 
a=the equatorial radius of the earth, and 
r=the radius of the parallel of latitude of which /¢ is an 
element, | 
we have, since the elemental ares ee’ and It are proportional to their 
respective radii a and 7, 


aM ee a 
dm lt a ° . . r e e e e e (1) 





as the fundamental equation, expressing the law of the Mercator 
projection. | 


5 


At any point on the spheroid, let 


p=the astronomical latitude, 

fP=the radius of curvature of the meridian section, 

p, =the radius of curvature of the section normal to the meridian, 
c=the compression of the earth, and 
e=the eccentricity of the meridian. 


By compression is meant the difference between the equatorial and 
polar radii, expressed in parts of the equatorial radius as unity, or 


cL a 


in which D=the polar radius. 
The eccentricity of the meridian is the distance of either focus from 
the centre, also expressed in parts of the equatorial radius as unity, or 


ea ati) 2 
a 
e=(2c—c’)? 
The values of the principal radii of curvature are— 
pP,=a (1—e’) (l—e? sin?p) 2 
and 
a 
p,=a (1—e’ sin? ~) ? 
Observing the figure, we have, as the length of ct, the length of an 
element of the meridian, 


se ff ai tere te Oe A Ce aE | 4, 


and, from equation (1), dM, the corresponding length of the chart me- 
ridian = - xdm; 
Therefore, 
dM=" . p, . dy ch secant Lally (3) 
From the figure, 
V=/,-COS P=AaCcOS Pp (1l—e’ sin? p) ? eer came oe, , (£) 


Substituting in (3), the value of pm, and the value of r from (4), we 
have 





_— al—é)dp . 
(arena cout etree) fay 2) 


Equation (5), by integration between the proper limits of latitude, 
will give the required length of any finite portion of the chart meridian, 
representirg the corresponding meridional arc on the terrestrial sphe- 
roid within the same limits. 


6 


For this integration, multiplying e’ in the numerator by sin? p+ cos? g, 
equation (5) may take the form 





dma“: 2? _v cos p. dp 
cos pm 1—e sin? p 


Again, multiplying the numerator and denominator of the first term 
by cos g, substituting 1—sin® ¢ for cos’ ~, and resolving both terms into 
partial fractions, the last expression becomes 








_a/( COS Pp COS P_\ 7,4? ( _€ COS P € COS M 
ae 1+sin mg 1—sin p ae sin @ Ta e sin ws ai 


For the length of that part of the meridian of the Mercator chart 
between the equator and any latitude, mg, we have 


we 
Mee COS @ COS Pp ) re € COS ~P € COS @ 
pee (ryan gt gp 1—sin p “s \Ge sin pp Bt ain P ae 


a LAE Sp ae rie sin @ 
=«(5 log l—sin pg 2 eo obee een Pp 











in which the logarithms belong to the hyperbolic system. 
Converting the logarithms in the foregoing expression to the common 
system, it becomes 





M=the meridional parts=a log, 10 log 5 I+sin p_1 ojo¢1+¢ sin p 
—sn gp 2 1l—e sin p 
Substituting for ma aie its trigonometrical equivalent, and for 


I+¢ SID @ its algebraic development, we obtain 
1—e sin Pp 





M=a log, 10 log tan( 459+ f )—a( esi g +56 sin® ptre sin? p+. -) 


which is the formula for computing the values given in the following 
table: 
COMPUTATION OF THE TABLE OF MERIDIONAL PARTS. 


The following numerical values, with their respective logarithms, have 
been employed for the several constants: 

1. The equatorial radius is expressed in minutes of are, in order that 
the unit of measure of the roeridional part M may become a minute of 
the equator, or a geographical mile. The value of a is, therefore, 


__360° x 60! 10800! rl b 
he SAG ENIRRGTR CS ee - + « « ». lOS== Seems 





2. The value of log, 10 is 2.30258509 . . . . log=0.3622157. 


7 


3. The eccentricity of the meridian is that based upon Clarke’s deter. 
mination of the compression in 1880, which is 


C= 5p gg 0.003407562 . . + + log=7,5824437, 
whence we have 
e=(2c—@)*=.08248146 . . . . log=8,9163666 
Hence the values of the several coefficients with their logarithms are 
a log. 10—7915'.7044558 . . . . log=3.8984895 
Ge ==) ano 11. oh eas LOS OG900T2 
4ae¢@ == 8 0’.053042 ees NLOP=RO 246192 
tae = £0/,000216523 . . . . log=6.3355038 


Accordingly the formula for practical use becomes 
M=7915.7044558 log tan (45°+ 5) 


— (23/.38871 sin p+0.053042 sin? p+0.000216523 sin? p+ . . . .) 


The computations have been made with this formula, aided by inter- 
polations, to four places of decimals, and checked throughout by first 
and second differences. 

Three places of decimals are retained in the table. as they are found 
necessary for the accurate projection of charts. 





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PFT” | FIO SPST | CEL” | FIL “GLLT | EZT° | LOO‘STLL | LTL* | OFO“SFOT | SOL* | TOO ’SLST | 60° | FEL‘ZIST | GRO" | GLE*LHPT | LLO® | LZS‘Z8ET | 690° | GST ‘STST | c90° | 62s “FSZT 
FEL’ | OLB OPST | E&T* | 18S °8LLT | Bat” | SHE OLE | ZIT | 826 Eh9T | ZOL’ | GGF'LLST | F60° | O€9"!ISE | S80° | O6Z ‘OFFI | 9L0° | ISP“I8ET | 690° | 980°LTET | T90° | S9T“EScT 
€FT°* | LOL “ShST | SEI* | StF LLLT | ZZT* | E28 GOLT | SIT” | 918 ‘SHOT | SOL” | L6E‘9LGT |-S60° | LEC ‘OTST | Ga0* | COS’ShFI | LLO’ | FLE“OSEST | 690° | LTO‘9ISE | G90" | 9OT“ZSZT 
FPL" | E8S PHST | GEL” | OTE “OLLT | GZL* | 10L‘8OLT | TIT’ | SOL ‘IFOT | ZOL* | G6 GLST | S60° | HHP'GOST | $80" | IZL PFET | 920° | 862 ‘6LET | 690° || SFE‘FIET | 190° | SFO ‘ISZT 
ShL" | OFF SFBT | GET” | PRL 'GLLT | TAL | O89 °LOLT | ZIT | 6G “OFT | GOL | E61 ‘PLGL | £60° | TES “80ST | 80° | 9EO'EFFI | LLO’ | 1ZS‘SLET | 690" | GLB “ETEL | GON" | £86 6FCT 
Eb | L6G GP8L | GEL” | GSO'FLLE | CSL° | 8Sh°9OL1 | SILT* | I8F GEST | ZOT’ | T60°SLST | G60" | SEZ “LOST | F80° | SSG IFFT | 929° | SHI “LLET | 80° TI8 “ZTE | 190° | G26 °8FSI 
ShT° | FST TPST | GEL* | Oc6°CLLT | TET* | LEE “COLT | TIT’ | OLE SENT | GOT’ | GSE ‘TLGT | ZEN | MOT “90ST | FRO* | 898‘°OFFL | 940° | 690‘9LEL | 690° | GPL IIST | T90° | T98°LFCT 
€FI° 1TO‘OF8ST | TEL* | G8L*ILLT | TZT* | 91Z"FOLT | TIT’ | GSZ"LEOL | GOL’ | L88OLST | $60° | SL0°COSL | 80° | F8L°6EFT-| 920° | S66 °PLZET | 890° | FLOOIST | 190° | 008 ‘9FZT 
€tL° | 898 8EST | GET* | LEO ‘OLZT | 1ZL* | S60"EOLT | TIT” | SFL ‘9E9T | TOL’ | 98L°69GT | Z60° | T86S0ST | F80° | OOL‘SEFL | C20" | BIB*SLEL | 90° | 909 GOEL | 190° | 6EL“SHZT 
GPL’ | 9GL°LEST | TEL* | 92S “GOLT | TZT* | FLE“IOLT | O11’ | 8E0 “SENT | TOL’ | S89°S9CT | 860° | 888°Z0SLE | F80° | OI9°LEPT | 9L0° | SFB °ZLET | 890° | 8ES°80ET | T90° | SL9FFCT 
THT" | GRSOEST | TET” | S6E“B9LT | OST | FSS OOLT | ITT* | Le6"EE9T | TOL’ | F8S “LOST | 260° | OGL ‘LOGT | 80° | ZES‘9EFT | 920° | OOL*IZEL | 890° | OLF°LOEL | 190° | LI9“EFZT 
Chr” «| EPP CEST | TEL” | F9B“LOLT | TUI° | C&L “669T | TTT” | 9I8 "CENT | TOT* | €8F 99SE | 260° | FOL OOGT | €80° | GhR'SEFI | GLO" | T69*OLET | 890° | GOF*90EL | 090° LSS “ZPCT 
CFL” (0€ “PEST | TEL” | EET "G9LT | OCT* | ETD 869T | OTL” | MOL *TE9T | TOT’ | 28E°S9CT | 260° | GI96EFT | E80° | N9E*FEFL | GLO’ | 9T9“G9ET | 190° | SEE°EOST | 190° | 96% TFSI 
Chl’ | GST EERT | TEL” | ZOO SOLT | OZL* | E6F'LGOT | OIL | OAS “OSOT | TOT’ | 18a ‘FOST | G6N* | OZS'S6FT | 80° | E8S°EEFT | SLO’ 1¥S “89ET | 890° | LOS ‘POET | 090° | SEF ‘OFZT 
TEL’ * | 810 CERT | OFT” | CLR °EOLT | OZT* | ELE “9GOT | OIT* | N8F'6Z9T | COT’ | ISL “SSSL | 260° | 8zFLEFL | ER0° | 00S 'ZEFT | GLO" | N9F"LOST | L90° | 00Z*EOST | 090° | OLE °6EZT 
TPL’ | LL8°OS8T | OST* | ShL COLT | OZL* | ESS “CE9T | OLL* | 9LE°SC9T | OOT* | 180 °Z9ST | T60° | LEE ‘96FT | 80° | LIL*1SFI | PLN" | G6E"998T | 890° GEL GOST | T90° | STE “8Esr 
TPL* | 964 °668T | O€T* | GIO ‘IOLT | GTT* | PEL ‘FEST | GOL’ | L9Z°LZ9T | OOL’ | T86 ‘09ST | 160° | 9FZ°E6FT | E80° | FRO‘OEFT | SLO° LTE “S9ET | L90° | S90 °TOST | 090° CGS “LECT 
Tf | S6G ‘8G8T | O€T* | SSP OOLT | OSL * | FIOEG9T | OIL’ | LET “9Z9T | QOL’ | 188 °6SST | 160° | SST FAPT | €80° | TS6‘8ZFT | SLO° G¥G POET | 290° 866 “666 | 090° C6L “9EZT 
T#T° | FSV LOxT | O8T* | CGE “GSLT | GIT’ | C68 “TLL | GOL’ | SFO 'SZ9T | NOL* | T8L ‘8ST | 160° | #90 SAFL | G80° | 698°LZFT | FLO" | SOT°E9ST | 290° | 186°86ZT | 690° | 9ST ‘SEZT 
OFT’ | PIE ‘978 | 66T* | E2a‘8CLT | 61T* | OLL‘*O69I | GOT* | GEE “EZ9T | COT’ | T89°ZSST | 160° | SZ6°I6FT | 780° | ZBL ‘9ZFT | SLO° €60‘Z9EL | 290° | F98°L6G1 | 090° | 9L0°FECL 
I¥I° ‘LL “Se8L | O€1° | €60°LELT | GIT’ | LE9689T | BOL” | O€8 °ZEOT | G60" | Z8G‘9CST | T60° | G88 °O6FT | S80° | FOL ‘SCFL | FLO’ | GIO'IOET | 990° | 862 ‘962 | 6C0' LTO “€82T 
OFT €E0 “FOST | 6ZT° | $96 CCL1 | GIT* | SES "BROT | GOL” | TZL“IZ9l |.00T* | T8F'SSST | 060° | C6L ‘68PT | Z80° | G29 'FZFL | FL0° | SFE GSET | 290° T&L “S6cI | 090° LS6 “1EZT 
OFT’ | $68 °228T | 62° | SES‘FSLT | SIL’ | OSh°LB9T | GOT” | ZI9‘OZOL | BGO' | E8B‘FEST | 160° | TOL ‘*88FT | 180° T¥S “SZtT | FL0° 1Z8 ‘8GE8T | 990° | S99°FECE | 6E0° | 868 ‘NEST 
6E1l° | PSL “IGBT | 84L* | LOL “ESLT | SIT° | ZOF‘989T | SOL” | FNS GIOT | 650° | F8C°EGST | 060° | TID“L8FL | 80° | 6Sh'SZbT | FLO° | ZEL“LGST ; 290° | 86G°S62zT , 6G0° | 6E8°6ZCT. 
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6EL° | SLb‘6IST | 8ScL* | OCF ISLT | STL’ | CQO°F8ST | SOT* | L8Z"LIOT | 660° | 980°ISST | 060° | TEP *ESPL | 280° | S6Z ‘OSI | FL0° | OS9"SCS1 | 990° | 99F°IGZT | 6C0° TIGL “LEST 
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GET" | OEE “BIST | RcI* | GEE OSLT | SIT’ | LEG “ZROT | 8OT* | GLT 9TOT | 860° | 886 ‘6ST | 680° | ZFE"FSFI | 180° | FIZ ‘GIFT | S20" LLG ‘FSET | 990° | OOF OGZL | 660° | 699 “9ZZT 
6EI° | L6L LIST | 8cT° | FET 6PLT | LIT’ | O€8°T89T | 801° | TLO“SI9T | 660° | 688 SPSL | 060° | ZGZ°ESFT | 180° SSI SIFT | 720° €0S “ESEL | 990° | FEE “682T | 60° £09 “SZEI 
6&T° | 890 9I8T | 8cL* | 990 SFLT | SIT" | GIL ‘OR9T | LOL* | F96E19NT | 860° | 16L“LHST | 680° | S9T°Z8FLT | T80° | ZGO LIFT | SLO" | OSR°ZSET | 90° | 69S‘RBGT | 8G0° | SHS *HZCT 
661° 616 ‘FIST | LOL* | GEG ‘SFLT | LILT® | S6S‘6LOT | SOL’ | 9S8°ZTOT | 860° | 69 ‘9FET | 060° | GLO ‘I8PI | 180° TL6STFT | €40° | LSE°IGEL | 990° G0Z “L8SCL | 890° | L8PhEcéT 
Sgt’ 18L S18 | 82T° | TI8 “SLT | LT’ | SLP'SLOL | LOT’ | GEL ITOT | 260° | S6S SFST | 680° | $86 6LPT | 180° | O68 FIFT | €L0° $86 ‘OGET | 90° SEL “98ZL | 6C0° | 86R ‘SSaT 
6EI° | GPO GTEL | Lol” | F89 FPLT } LIL* | 196 °LL91 | LOT | SF9 OTOL | 860° | LEF'FFST | 680° | S68 ‘8ZFT | TS0° 608 “STFL | ZL0° GIG ‘6PEL | 990° | ZLO'S82T | 8S0° | OLE *TZZT 
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Lele LOE “OTST | LOT” | OP CHLT | LIT’ | LET SLOT | LOT” | 828091 | 860° | TOE *ZFHST | 680° | LIL ’OLPL | T8O° | SPO "TIFT | €20° | 990°LHEL | G9N° | ZHE°S8ZT ;, 8C0° | ESS ‘6IZT 
S&T’ 6c “608T | LEL* | COE THAT | OIT* | 1TO°FLOL | LZOL® | 1ZE°L09T | 260° | FOZ IFST | 880° | 629 °CLPT | 080° 89¢ ‘OIFL | 210° 66 “CFET | C90° LL8 “T8eL | 820° C6T SIZT 
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